Relay on-off threshold for NDF protocol with distributed orthogonal space-time block codes

In this article, a relay on-off threshold (ROT) based on symbol error rate is derived for the cooperative communication networks with multiple antennas, where the non-orthogonal decode-and-forward (NDF) protocol with source antenna switching and linear combining decoding are used as the relaying protocol and decoding scheme, respectively. The optimal ROT for the cases using distributed orthogonal space-time block codes is derived for high signal to noise ratio (SNR) region and also a suboptimal ROT is provided for low SNR region. Finally, the diversity order of the NDF protocol using the relay on-off scheme with the proposed ROT is derived and its performance is verified through numerical analysis.

Attenuation and fading in the multipath wireless environment make it difficult for a receiver to correctly decode a received signal. Therefore, replicas of a transmitted signal are sent to the receiver to improve the detection performance.

Tarokh et al. [1] proposed space-time codes (STCs) to achieve transmit diversity and their design criteria. Alamouti code was proposed in [2], which was extended to orthogonal space-time block codes (OSTBCs) in [3]. Also, the symbol error rate (SER) and bit error rate (BER) of OSTBCs for quadrature amplitude modulation (QAM) were derived in [4,5], respectively.

The cooperative diversity which is obtained by utilizing relays between source and destination has been actively studied since Cover and El Gamal’s [6] work. Laneman et al. [7] developed some cooperative diversity schemes based on repetition codes and analyzed their performance, and Wang et al. [8] developed a good demodulation scheme for the protocol proposed in [7].

Recently, various results on how to select the best relay(s) among multiple relays have been reported. Bletsas et al. [9] proposed an opportunistic relaying which selects a relay only by using local channel information. Jing and Jafarkhani [10] proposed a multiple-relay selection scheme for the amplify and forward (AF) protocol and Yi and Kim [11] analyzed the diversity order of the decode-and-forward (DF) protocol with relay selection.

On the other hand, a relay on-off scheme for the DF protocol according to the channel state was proposed in [7]. Recently, another relay selection scheme based on the relay on-off threshold (ROT) for the DF protocol was also proposed in [12]. In both schemes, if the received signal to noise ratio (SNR) at the relay is larger than the ROT, the relay transmits a signal and otherwise, the relay does not transmit a signal. However, the optimal ROT was not derived analytically.

For the orthogonal DF (ODF) protocol and binary phase shift keying (BPSK), Siriwongpairat et al. [13] derived the ROT over the Rayleigh fading channel and Ikki and Ahamed [14] derived the ROT over the Nakagami-m fading channel. However, until now, for M-QAM and non-orthogonal DF (NDF) protocol in which a source also transmits a signal when a relay transmits a signal, the ROT has not been investigated.

It is well known that source antenna switching (SAS) can gain additional diversity for the NDF protocol by using different antennas at the source in the first and second phases [15], and the maximal likelihood (ML) decoding is impractical because its computational complexity increases exponentially with the signal constellation size. Therefore, we will use the low-complexity decoding scheme for OSTBCs given in [16,17], which is called linear combining (LC) decoding.

In this article, an ROT is derived for the NDF protocol with distributed orthogonal space-time block codes (DOSTBCs) and M-QAM such that the SER of LC decoding is minimized over the Rayleigh fading channel. Until now, this kind of ROT has not been investigated yet. And, the diversity order of the proposed relay on-off scheme is derived.

This article is organized as follows. In Section “System models and LC decoding”, the system models and LC decoding are described and in Section “ROT and diversity analysis”, the ROT based on SER is derived and the diversity order of the proposed relay on-off scheme is analyzed. In Section “Numerical analysis”, the SER performance of the proposed relay on-off scheme is evaluated through numerical analysis and the concluding remarks are given in Section “Conclusion”.

Throughout this article, the following notations are used. E[·] denotes the expectation. X∼CN(0,σ²) denotes the complex Gaussian random variable with zero mean and variance σ²/2 for each of real and imaginary parts. For a complex number, |·|, (·)^◦, , and denote the magnitude, the complex conjugate, the real part, and the imaginary part, respectively. and (·)^T denote the set of M×N complex matrices and the transpose of a matrix, respectively.

For the three-terminal cooperative communication network in Figure 1, it is assumed that source (S), relay (R), and destination (D) can have multiple antennas and the channels are quasi-static Rayleigh fading ones. Since non-orthogonal transmission is assumed, both source and relay transmit signals in the second phase. It is also assumed that the destination has the perfect channel state information (CSI) of S–D and R–D channels and the relay knows the CSI of S–R channel and the mean and variance of S–D and R–D channel coefficients. For simplicity, Gray-mapped M-QAM is assumed.

If the SNR at the relay is larger than an ROT T in the DF protocol, the relay transmits a signal and otherwise, the relay does not transmit a signal. Note that it is assumed that the source and destination know whether the relay is on or off. In the cooperative communication network, if the source uses the same antenna in the first and second phases, the SD and SR channel coefficients for these two phases become identical. Therefore, in order to increase the diversity order, it is assumed that the SAS [15] is used.

System models of NDF protocol

It is assumed that the numbers of antennas of the source, relay, and destination are M_S, M_R, and M_D, respectively. Let x be the message vector consisting of L independent symbols. denotes the codeword of OSTBC transmitted from the source, where denotes the number of transmissions at each source antenna. In the first phase, the received signal matrices Y_R and Y_D1 at the relay and destination can be written as

where and are the S–R and S–D channel matrices, and and are the noise matrices at the relay and destination. Note that p₁ρ is the power of the signal from the source and ρ is a parameter linearly proportional to the average transmit SNR.

In the second phase, let x^R denote the decoded message vector at the relay and and denote the codewords of OSTBCs corresponding to x and x^R transmitted from the source and relay, respectively. When the relay is on, the received signal at the destination in the second phase is given as

where and are the S–D and R–D channel matrices, and is the noise matrix at the destination. Note that p₂ρ and p₃ρ denote the signal powers from the source and relay in the second phase, respectively. Then also forms an OSTBC if x^R=x. The entries of H, G₁, G₂, and F are independently distributed as , , , and , respectively. The entries of N_R, N_D,1, and N_D,2 are i.i.d. random variables distributed as CN(0,1).

When the relay is off, the received signal at the destination in the second phase is given as

Therefore, by using (2), (3), and (4), the received signal matrix can be expressed as

LC decoding

The optimal ML decoding at the destination is done by selecting such as

Since, the complexity of ML decoding increases exponentially to the constellation size and the analysis of ML decoding is very difficult, simple and practical LC decoding [16,17] will be considered in this article. It is clear that LC decoding is equivalent to ML decoding for OSTBCs but it is not true for DOSTBCs because the relay may transmit erroneously decoded symbols.

For the cooperative communication network, the LC decoding operates twice. First, the LC decoding is done for Y_D1 and Y_D2, respectively, before the decision on x is made at the destination. Then, the destination combines the decoder outputs for Y_D1 and Y_D2. Finally, the decision is made for x by using this combined output.

If the SR channel or RD channel is not good enough for the reliable communication, a relay cannot help the destination that much. Therefore, it is important to decide whether a relay should be used or not depending on the states of SR and RD channels. In general, a relay is used in the DF protocol only when the received SNR at the relay is greater than the ROT. If the relay correctly decodes the information, it can be thought as a virtual antenna of the source and the diversity gain can be obtained even though the RD channel is not good. Therefore, an ROT can be determined only by monitoring SR channel state [7,13,14].

If the ROT is too low, the relay may transmit many erroneous data under the bad S–R channel condition, which causes many decoding errors at the destination. On the contrary, if the ROT is too high, the relay is rarely used and the cooperative diversity cannot be achieved. Therefore, the optimal ROT for the NDF protocol with DOSTBCs and M-QAM is derived in this section.

It can be assumed that, in high SNR region, only one symbol error occurs among L symbols in a codeword and further a symbol error is caused by one-bit error. Let , , , and , where g_k,i,j denotes the channel coefficient between the ith source antenna and the jth destination antenna in the kth phase, and h_i,j denotes the channel coefficient between the ith source antenna and the jth relay antenna, and f_i,j denotes the channel coefficient between the ith relay antenna and the jth destination antenna. Clearly, u, g₁, g₂, and w are Erlang distributed [18].

ROT (general expression)

To derive the ROT minimizing the SER, the SER should be derived first. For the NDF protocol using a relay on-off scheme, the symbol error event at the destination can be divided into three cases. The first case is that symbol error occurs at the destination when the relay is off. The second case is that symbol error does not occur at the relay but symbol error occurs at the destination when the relay is on. The third case is that symbol error occurs at both the relay and destination when the relay is on. Therefore, the SER P_e(T) with the ROT T at the destination is expressed as

As SNR increases, the probability of Case 3 in (6) can be approximated as

The pdf of Erlang random variable u is given as

where . Let P_e1 be the SER at the destination when the relay is off, P_e2 be the SER at the destination when the relay is on and no decoding error occurs at the relay, and P_e3 be the SER at the destination when the relay is on and the decoding error occurs at the relay. Using (6) and (7), the SER at the destination is derived as

where P_e3 can be approximated as the symbol error probability at the destination when one bit error causes one symbol error at the relay. Then, by solving for (9), the ROT to minimize (9) is given as

Since some approximations are used to derive the ROT, it is clearly suboptimal. However, through simulation, it will be shown that this ROT approaches the optimal ROT as SNR increases.

ROT for LC decoding

In this section, we derive P_e1, P_e2, and P_e3 for LC decoding and obtain the ROT based on them.

Case (1) Relay is off:

Since only the source transmits signals, P_e1 can be written as [4]

where d denotes the distance between one symbol point and the adjacent decision boundary in the rectangular QAM and the closed form of P_e1 can be derived by the result in [19].

Case (2) Relay is on and no decoding error occurs at the relay:

Since the relay transmits correct data, P_e2 can be written as [4]

and its closed form can be derived by the result in [19].

Case (3) Relay is on and the decoding error occurs at the relay:

Since Gray-mapped M-QAM is assumed, the SER is dominantly determined by the single-bit error at the relay as SNR increases. Also, the most frequent symbol error event at the relay is one symbol error among L symbols. Therefore, P_e3 can be approximated as the SER at the destination when one-bit error occurs at the relay. Since the SER depends on the LC decoder output before the decision is made, we confirm how the one bit error at the relay affects the LC decoder output at the destination and then derive the SER.

When the relay erroneously decodes x_k to x_k + 2d, the LC decoder output s_k for x_k at the destination can be written as

Next, let us investigate the LC decoder output s_i,i≠k for the x_i which is not erroneously decoded at the relay. For OSTBCs, when the LC decoding is performed to the data symbol x_i, the terms related to the other symbols are canceled out and thus the decoder output at the destination becomes equivalent to (14). However, for DOSTBCs, if one bit error for x_k occurs at the relay, the terms related to x_k possibly affect the decoder output for the other symbol at the destination. Therefore, the decoder output for x_i at the destination can be divided into two cases. First, the terms related to x_k do not affect s_i such as

Second, the terms related to x_k affect s_i due to one bit error for x_k at the relay such as

where n_i is distributed as CN(0,p₁g₁ + p₂g₂ + p₃w), g_2,i,j∘f_l,k denotes either or depending on the used DOSTBC, and D is the set of indices of channel coefficients that are not canceled out due to one bit error for x_k.

As an example, consider a DOSTBC using the following OSTBCs for M_S=2 and M_R=2

If an error occurs for x₁ at the relay, s_2,R and s_2,I are given as (15) and s_3,R and s_3,I are given as (16). If an error occurs for x₂ at the relay, s_1,R and s_1,I are given as (15) and s_3,R and s_3,I are given as (16). If an error occurs for x₃ at the relay, s_1,R, s_1,I, s_2,R, and s_2,I are given as (16).

From the LC decoder outputs (13)–(16), we can check the error cases at the destination and then calculate P_e3.

The SER for x_k (the error symbol at the relay) at the destination;First, we consider the case that the error occurs at the destination when decoding s_k in (13) and (14). Note that k is the index of the error symbol at the relay. Note that the relay can erroneously decodes x_k to x_k−2d or x_k±j2d. However, the identical LC decoder outputs to (13) and (14) are obtained since the rectangular QAM is considered. Thus, for those cases, we can also obtain and P_e3,2 similarly. Therefore, when one bit error occurs for x_k at the relay, the SER for x_k at the destination is given as the linear combination of and P_e3,2, that is

where β and γ_k are constants determined by the constellation and structure of DOSTBCs. The coefficient of P_e3,1 in (21) is 1 because P_e3,1 can be regarded as the probability that the same error at the relay also occurs at the destination.

The error case that the destination decodes to From (13), if and there is an adjacent symbol of x_k in this direction, the destination can erroneously decode to with the following SER

Note that this symbol error at the destination is the same as the symbol error at the relay.

The error case that the destination decodes to From (13), if , this symbol error for can also occur with the following SER

The error case that the destination decodes to From (14), if or , a symbol error for can occur at the destination with the following SER

The SER for x_i,i≠k (no-error symbol at the relay);The SER for x_i can be obtained according to the LC decoder output in (15) and (16). Finally, the SER for x_i,i≠k can be expressed as

where γ_i and δ_i are constants determined by the constellation and structure of DOSTBCs. Since the LC decoder output for x_i cannot become (15) and (16) simultaneously, for each P_error,i, if γ_i is not zero, δ_i should be zero and if δ_i is not zero, γ_i should be zero.

The case that LC decoder output for x_i is (15)From (15), if or , the destination can erroneously decode to or . Also, if or , the destination can erroneously decode to or . In fact, the SER for all these cases is the same as P_e3,2.

The case that LC decoder output for x_i is (16)Similarly to the previous case, we can obtain the following SER by considering (16)

where denotes one of , , , and depending on the structure of DOSTBCs like the operator ∘ in (16). However, all possible P_e3,3’s depending on ⊕ have the same value because each channel coefficient is a complex Gaussian random variable with zero mean. Thus, we can see that ⊕ can be replaced with the addition without changing the result.

The SER at the destination when one bit error occurs at the relay (P_e3);When one-bit error for x_k occurs at the relay, the SER at the destination can be expressed as

By plugging P_e1, P_e2, and P_e3 to (10), we can obtain the ROT. In fact, we cannot derive or prove something not because it is complicated but because it is hard or intractable. However, the ROT for high SNR region can be derived by the following approximation.

Since the direct approximation of P_e3,1 from (18) is too complicated, we use the assumption that the effect of noise is negligible in high SNR region and thus (13) is approximated as

Then, the error can occur if in high SNR region and thus the SER (P_e3,1) is given as

By using the pdf and the cumulative distribution function (cdf) of the sum of independent and nonidentical Erlang random variables in [18], can be easily calculated.

As SNR increases, , and P_e3,3 in (24) become negligible compared to and thus P_e3 becomes close to . Since P_e2 in (12) decreases much faster than P_e1 in (11) as SNR increases, only P_e1 and become dominant in the numerator and denominator in (10), respectively. Finally, in high SNR region, the ROT in (10) is simplified as

Decision of suboptimal ROT in low SNR region

In the previous section, an ROT was derived only for high SNR region because P_e3 cannot be derived in low SNR region. Thus, for the whole SNR region including low SNR region, the ROT should be determined.

The ROT in (10) is approximated as (27) by using the following approximation

However, the ROT in (27) cannot be calculated in low SNR region because becomes less than P_e1 as SNR decreases. The argument of Q⁻¹(·) can be greater than 1 when the R–D channel state is not better than the S–D channel state such as (ρ,ρ,ρ). Therefore, a suboptimal ROT for low SNR region is heuristically obtained by giving a bias such as using instead of in (28). Thus the ROT in (27) can be modified as

Note that, as SNR increases, the ROT in (29) becomes identical to the ROT in (27) because P_e1 goes to zero.

Diversity analysis

In this section, the diversity order of NDF protocol with the proposed relay on-off scheme is derived when DOSTBCs and LC decoding are used.

When the relay is off, the transmit signals in the first and second phases form an OSTBC and when the relay is on, the transmit signals in the first and second phases form a DOSTBC .

Since the ranks of their difference matrices are 2M_S and 2M_S + M_R, respectively, and the destination has M_D antennas, in high SNR region, P_e1 and P_e2 are proportional to ρ^−2M_SM_D and ρ^−{M_SM_D + (M_S + M_R)M_D}, respectively. Also, in (27) can be approximated as cρ^−2M_SM_D, where c is a positive constant.

Since , where [20] in high SNR region, the ROT in (27) can be approximated as

For simplicity, let and g(z) be the pdf of z. Then the SER in (9) at the destination is rewritten as

Now, we derive the diveristy order for each term in (31). Since , the cdf of z is given as

where .

The first term in (31) can be given as

where (λT / ρ)^M_SM_RP_e1/(M_SM_R)! is dominant because T/ρ < < 1 as ρ increases. Using (30) and (33), the diversity order of the first term in (31) is obtained as 2M_SM_D + M_SM_R.

As SNR increases, goes to zero and thus becomes dominant in the second term in (31). Therefore, it is clear that the diversity order of the second term in (31) is 2M_SM_D + M_RM_D because .

Let and δ=3/(M − 1). Then, the third term in (31) is upper bounded by

Note that the factor of the first term in (34) becomes dominant as ρ increases. By plugging (30) into (34), this factor becomes as follows.

Since in becomes negligible as SNR increases, the diversity order of (34) is given as 2M_SM_D + M_SM_R.

In conclusion, the diversity order d_LC of NDF protocol with the LC decoding and the proposed relay on-off scheme is given as

Note that the diversity order M_RM_D) is identical to the full diversity order of NDF protocol with SAS [15].

For the simulation, it is assumed that the transmit signal power in the first phase is the same as the sum of transmit signal powers from the source and relay in the second phase, and the transmit signal power of the relay is the same as that of the source in the second phase. (x,y,z) denotes the average received SNRs in dB of S–D, S–R, and R–D channels, respectively. For example, (ρ,ρ,ρ + 6)means that the average received SNR of R–D channel is larger than those of S–R and S–D channels by 6 dB. For all simulations, 16QAM is used.

To confirm the diversity order, when M_S,M_R,M_D≥2, simulation must be performed in very high SNR region, which requires too long time. Instead, simulation has been performed for the single-antenna case (M_S=M_R=M_D=1). For M_S=M_R=M_D=1, the transmitted OSTBCs are given as

where the combination of C_S,2 and C_R,2 forms an Alamouti code in the second phase. The ROT for the single-antenna case can also be easily obtained from (27) and (29).

Figures 2 and 3 compare the performance of NDF protocol with various relay schemes. We do not consider the case that the S–R channel state is better than the other channel states because the relay always tends to be on. The performance of the proposed optimal and suboptimal relay on-off schemes and the conventional relay scheme are denoted by ‘optimal-relay-on-off’, ‘subopt-relay-on-off’, and ‘relay-on’, respectively. In other words, ‘relay-on’ means that the relay always transmits signal. The ‘direct transmission’ implies that the relay is always off. The optimal relay on-off scheme uses the optimal ROT which is determined by extensive simulation and the suboptimal relay on-off scheme uses the ROT in (29).

Figure 2 shows that the SER performance of the relay-on scheme becomes worse as the R–D channel state becomes better and thus, in this case, the relay on-off scheme is vital to improve the LC decoding performance. For the case of (ρ + 6,ρ,ρ) in Figure 3, it can be seen that the performance of the relay on-off scheme is almost identical to that of the direct transmission until ρ=12.5 dB, which is because P_e1≤P_e2 and the relay is always off. Therefore, the ROT should be infinite, as given by (10). Also, we can see that the analytical diversity results in Section Diversity analysis are well matched with the simulation results in Figures 2 and 3, and the suboptimal ROT works well in low SNR region as well as in high SNR region.

Figures 4 and 5 compare the performance of NDF protocol with various relay schemes when M_S=M_R=M_D=2 and the DOSTBC in (17) is used. In Figure 4, since the diversity effect appears gradually from ρ=12.5 dB, it can be expected that the diversity order of relay on-off scheme will reach 12 in higher SNR region. For the case of (ρ + 6,ρ,ρ) in Figure 5, it can be seen that the performance of the relay on-off scheme is almost identical to that of the direct transmission, which implies that the ROT is infinite. Since the diversity effect will appear as SNR increases, it is expected that the relay becomes helpful as a virtual antenna to increase the diversity order in higher SNR region in this case. Also, the simulation results show that the suboptimal ROT works well in low SNR region as well as in high SNR region.

In this article, an ROT was analytically derived for NDF protocol with DOSTBCs and SAS in high SNR region, where LC decoding was considered because it has low complexity and good performance. Through the diversity analysis, it was confirmed that the NDF protocol with the proposed relay on-off scheme can achieve the full diversity. For low SNR region, the suboptimal ROT was provided and the simulation results confirmed that this suboptimal ROT works well in whole SNR region. It is left as a future work to derive the optimal ROT for the cooperative communication networks with multiple relays by using a similar method to one proposed in this article.

The authors declare that they have no competing interests.

This research was supported by the KCC (Korea Communications Commission), Korea, Under the R&D program supervised by the KCA (Korea Communications Agency) (KCA-2012-08-911-04-003) and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2012-0000186).

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